A378931 Triangle read by rows, based on products of Jacobsthal numbers (A001045).

1, -1, 3, -2, -9, 15, -4, -18, -25, 55, -8, -36, -50, -121, 231, -16, -72, -100, -242, -441, 903, -32, -144, -200, -484, -882, -1849, 3655, -64, -288, -400, -968, -1764, -3698, -7225, 14535, -128, -576, -800, -1936, -3528, -7396, -14450, -29241, 58311, -256, -1152, -1600, -3872, -7056, -14792, -28900, -58482, -116281, 232903

Offset: 1

Author

Werner Schulte, Dec 11 2024

Keywords

• sign
• easy
• tabl

Comments

Let M = T^(-1) be matrix inverse of T seen as a lower triangular matrix. M is a harmonic triangle with M(n, k) = 1 / A084175(n) if k = n, and 1 / A378676(k) if 1 <= k < n. Triangle M(n, k) for 1 <= k <= n starts:

1/1

1/3 1/3

1/3 1/5 1/15

1/3 1/5 1/33 1/55

1/3 1/5 1/33 1/105 1/231

1/3 1/5 1/33 1/105 1/473 1/903

etc.

Sum_{k=1..n} M(n, k) * 2^(k-1) = 1.

Sum_{k=1..n} M(n, k) * (-2)^(k-1) = (-1)^(n-1) / A001045(n+1).

Sum_{k=1..n} 2^(k-1) / M(n, k) = (8^n - 1) / 7 = A023001(n).

Examples

			Triangle T(n, k) for 1 <= k <= n starts:
n\k :     1     2     3      4      5      6       7       8      9
===================================================================
  1 :     1
  2 :    -1     3
  3 :    -2    -9    15
  4 :    -4   -18   -25     55
  5 :    -8   -36   -50   -121    231
  6 :   -16   -72  -100   -242   -441    903
  7 :   -32  -144  -200   -484   -882  -1849    3655
  8 :   -64  -288  -400   -968  -1764  -3698   -7225   14535
  9 :  -128  -576  -800  -1936  -3528  -7396  -14450  -29241  58311
  etc.
		

Crossrefs

Cf. A001045, A023001, A378676.

A084175 (main diagonal), A139818 (1st subdiagonal), A000079 (column 1 and row sums).

Programs

Mathematica

T[n_,k_]:=If[k==n, (2*4^n-(-2)^n-1)/9, -2^(n-1-k)*(2^(k+1)+(-1)^k)^2/9]; Table[T[n,k],{n,10},{k,n}]//Flatten (* Stefano Spezia, Dec 11 2024 *)

PARI

T(n,k)=if(k==n,(2*4^n-(-2)^n-1)/9,-2^(n-1-k)*(2^(k+1)+(-1)^k)^2/9)

Formula

T(n, n) = (2 * 4^n - (-2)^n - 1) / 9 = A084175(n), and T(n, k) = -2^(n-1-k) * (2^(k+1) + (-1)^k)^2 / 9 for 1 <= k < n.

G.f.: x*t * (1 - 3*t - 6*x*t^2 + 8*x^2*t^3) / ((1 - 2*t) * (1 - x*t) * (1 + 2*x*t) * (1 - 4*x*t)).